Week 6: What are the challenges in shifting content from “what” to “where” and “how”?

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In mathematics education, it has always been a struggle for me to utilize the upper levels of Bloom’s Taxonomy. In my early years of learning and teaching, there seemed to be a major focus on the lower levels, and not much emphasis on the upper levels. Writing in mathematics was almost non-existent. In fact, most of my current students question the practice of writing complete sentences in math assignments, even if current texts provide opportunities to do so. The expectation of my students is to provide numerical responses, along with necessary “showing work”, but the idea of writing or providing other levels of processes in Bloom’s Taxonomy is considered extraneous and superfluous.

Fortunately, most current textbook publishers provide teachers with resources to provide great opportunities to utilize all the levels in practice problems, traditional assessments, and performance assessments. For example:

A teacher gives a test to a class of 25 students. The grades are as follows: 40% of students receive a grade of C (test scores of 70% to 79%), 40% of students receive a grade of either B (80% to 89%) or D (60% to 69%), 20% of students receive a grade of either A (90% to 100%) or F (0% to 59%).

a) Write a set of 25 test scores based on the information above.

b) From your data, determine the mean, median, and mode for the class.

c) Draw a graph that properly displays your data.

Demonstrate your knowledge by designing a water tower tank with the following requirements. The height of the water tank must be 30 feet. The horizontal tank dimensions must cover as much as possible of the 70-foot by 70-foot plot of land that the tower will sit on. The tank must have a volume of at least 90,000 cubic feet and not more than 100,000 cubic feet. The potential choices for the shape of the tank are cone, cylinder, pyramid, prism, sphere, and hemisphere.

1. On a separate sheet of paper, draw each figure based on the height and land requirements. Label the dimensions for each.

2. Based on the requirements in Exercise 1, find the volume for each figure to the nearest cubic foot.

3. Describe any shortcuts you could take in finding the volumes.

4. Which shapes meet the water tower tank volume requirements? Explain how the figures in Exercise 1 can be modified to fit the tank’s volume requirements. Describe the new dimensions and volume for each to the nearest tenth cubic foot.

These examples show that they ask students more than just remember or understand processes.

The National Council of Teachers of Mathematics (NCTM) has encouraged this higher level thinking as well. Mathematical modeling has been a strategy to teach meaningful learning of mathematics content. It is “a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena” (Hernandez et al., 2017). It has been a struggle for high school math teachers like me to implement this type of learning process. Hernandez describes how you can introduce this mathematical modeling in your class. (2017)

Start small

Scaffold initial experiences with leading questions and class discussion

Use common, everyday experiences to motivate the use of mathematics

Use bite-sized modeling scenarios that require only one or two components of a full modeling cycle

Share your goals and instructional practices with parents and administrators

I think it would benefit all students as they learn math content to be provided real-world problems to solve. Many math texts, in my opinion, provide what I call story problems for the sake of making a story problem, which is not real-world. The issue is that in order to solve real-world problems, students need a full skill set of math tools, strategies, and content to do this, which usually is encountered later in their secondary math career. It is the goal of all math teachers to provide students with mathematical modeling opportunities to solve problems that will arise in their lifetime and hopefully make a difference.

9 thoughts on “Week 6: What are the challenges in shifting content from “what” to “where” and “how”?”

I also struggle with using the higher levels of the taxonomy, especially in chemistry. So much of the chemistry curriculum seems to be about defining terms, when there is so much more that we could do. I liked the example problems that you shared because I think I could do something like that with my chemistry students, specifically in the second semester when we get into the material that better lends itself to problem solving. I think I could also use the steps you shared for using mathematical modeling to start using more problem-based questions for my students in chemistry. Thanks for the great resources! If I can use more real-world problems in chemistry I can imagine students would not only be more engaged, but probably also start performing better.

You would think it’s easy to use real-world problems in school, but we have focused so much in the last 15 years on standardized testing, that we have totally focused on the “what” and nothing else! It’s frustrating. I think chemistry would be an awesome area to make real-world problems. There is chemistry everywhere! Good luck…

I also wrote about Bloom’s Taxonomy for this week. I like how you used some of your math references in your blog. I can see how using the higher level thinking could be challenging. I know when I used the textbook problem solving, I was always changing the names and locations to better suit the students thinking.
Do you or have you considered using math journals with your students? I found this beginner’s approach source @ http://www.nctm.org/Publications/mathematics-teacher/2000/Vol93/Issue2/mt2000-02-132a_pdf/
I enjoyed using them with students in primary school. It really focuses on the process, reflective thinking, vocabulary and developing problem solving skills.

I have heard of having students write in journals, but have never tried it. I think I need to be more assertive in this aspect. Math writing is very important. It allows students the opportunity to use the upper level processes of Bloom. As it stands, there are more writing opportunities in my Statistics class then in my Geometry class! And it’s mostly due to the fact that the assignments in stats class are written to have students respond in writing. They need to explain, justify, and interpret many things in the class, plus the fact that most problems in stats class are basically story problems.

So I just read your blog from your other class (didn’t realize both were on the same page) and wrote a very long comment on it because it related to playing games and I figured it was the “how” since that relates to playing and using your imagination. So I’m just going to leave the comment I wrote on the other one, even though it may not totally relate to the blog for this class, I think it relates in general to the “how” of learning.

I wish I had the use of computers on a regular basis at school to allow students to play games, unfortunately I do not, but have in the past so I personally know how beneficial they can be for students. A few years ago, I taught 8th grade math and science back to back (same students). I started to realize that there were quite a few students who were low in their basic math skills. I found a website called mangahigh.com, which used to have a lot more free games then they do now, and once or twice a week we would skip science and play math games throughout the period. It was cool to see kids get interested in the game and compete to see who could get a higher score without realizing they were actually doing anything math related. It allowed me to see an improvement in basic math skills in these students while allowing them to do something they enjoyed (or maybe they just wanted to get out of science, either way they were improving so I didn’t care).

Last year, the school I taught at was able to get a class set of tablets that we loaded a lot of math apps onto. I taught a combination 4th and 5th grade math and science class. A lot of times students either had a review or some students would get done with their math assignments way before we would transition into science. Having the tablets allowed them to play games during their free time. A lot of the kids got a lot quicker on their multiplication facts through these apps which was great. We also had a lot of legos in the classroom and other days they were allowed to create and use their imaginations, something else that I thought was very important to integrate into the math and science subjects.

I like the idea of stations in the classroom. I have used them a few times this year and the students really enjoyed them, even when there was no no technology incorporated. We were learning about angles and I had them write their names out and they had to label all the angles, vertexes, and highlight the adjacent angles. That was the kids favorite station because although they were doing math, they didn’t really feel like it was math. Another time we had a competition to see which groups could answer the most problems involving integers and whatever team won by the end of the period got a prize. Nowadays with websites with Pinterest, creating stations is a breeze because there are so many ideas that you can take and use in the classroom. Anytime I can make learning into stations or a game I know students enjoy it and learn a lot more than just listening to me talk (although we will never fully get away from some type of direct instruction in my opinion).

Thanks for the long response! I am very lucky to have a class set of Chromebooks. They are showing some wear, but they operate just great. Unfortunately, the class that utilizes them the most are my Algebra 1 Support class. My statistics class uses them on occasion, and my Geometry class has never used them! You are right about games, and how some kids think they aren’t learning. If that is happening, it’s great. There are definitely many resources out there on the internet, you just have to find it. And you are right, physical manipulatives are great too. We need to be diligent and have students use as many tools as they can to learn.

I think making real-world connections for lower grade math is much easier than higher grades. There are all kids of connections I can make with 5th graders when we talk about decimals and fractions and percentages. “Yes, you really are going to use this for the rest of your life.” I can’t think of very many real-world applications for Calc. or Trig. outside a mathematician or scientist or engineers. For a high school student making that connection is damn hard. And even harder for a teacher to come up with a realistic scenarios.
I read your comment on someone else’s blog about only focussing on a concept that is required per grade level. I like that idea. Leave the higher level math for later and when students choose to enter that field. Sometimes I think we try to cover way to much and don’t offer the kids a legitimate chance at becoming proficient at something before moving on. I understand spiraling and how it’s supposed to work, but from my experiences, it doesn’t work very well.

Your blog entry for this week reminds me of our new math curriculum. I may be one of the few teachers that actually likes it, with the exception of the pacing. In the 3rd grade curriculum there is a huge emphasis on the higher levels of Bloom’s Taxonomy. Students are no longer solving the lower level of Bloom’s Taxonomy scale. Some examples of how they are required to use a deeper level of understanding include questions such as: EXPLAIN your work/findings, Can you solve this problem another way?, If “so and so” did the problem incorrectly how can they fix their mistake?, There are so many word problems in the new curriculum, but it is helping students become better problem solvers. In the beginning this was a real struggle, but students are able to understand the “how” and “why”.

We need more teachers like you!!! I don’t think elementary teacher know the effect they have on early math learners. It’s great to know we have adopted a good curriculum that emphasizes higher levels of Bloom’s Taxonomy. It IS a struggle to have students come from recall mindset to one where they need to do more. But it is the right direction for math education. I appreciate the comment!